We study the problem of constructing infinite words having a prescribed finite set P of palindromes. We first establish that the language of all words with palindromic factors in P is rational. As a consequence we derive that there exists, with some additional mild condition, infinite words having P as palindromic factors. We prove that there exist periodic words having the maximum number of palindromes as in the case of Sturmian words, by providing a simple and easy to check condition. Asymmetric words, those that are not the product of two palindromes, play a fundamental role and an enumeration is provided.
Given a polyomino, we prove that we can decide whether translated copies of the polyomino can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider. If such a tiling exists the polyomino is called an exact polyomino. Further, every such tiling of the plane by translated copies of the polyomino is half-periodic. Moreover, all the possible surroundings of an exact polyomino are described in a simple way. 586 D. Beauquier and M. Nivat So Lemmas 4.1 and 4.2 have a corollary: Corollary 4.1. If (A, C, B') and (A, D, B') are two different contacts, then p is a pseudoparallelogram ABA'B'. We now formulate three technical lemmas about pseudoparallelograms which we shall need later.
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